DEM Prediction

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    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2008; 58:319353Published online 7 January 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.1728

    DEM prediction of particle flows in grinding processes

    P. W. Cleary1,,, M. D. Sinnott1 and R. D. Morrison2

    1CSIRO Mathematical and Information Sciences, Private Bag 33, Clayton, Vic. 3168, Australia2Julius Kruttschnitt Mineral Research Center, The University of Queensland, Isles Rd, Indooroopilly,

    Qld. 4068, Australia

    SUMMARY

    Particle size reduction is a critical unit process in many industries including mineral processing, cement,food processing, pigments and industrial minerals and pharmaceuticals. The aim is to take large feedmaterial and as efficiently as possible reduce the size of the particles to a target size range. Over time, avery large range of equipment has been developed to perform this for many materials and in many differentconditions. Discrete element method (DEM) modelling is a computational tool that can allow detailedexploration of the particle flow and breakage processes within comminution equipment and can assist indeveloping a clearer and more comprehensive understanding of the detailed processes occurring within.In this paper, we examine the particle and energy flows in representative examples of the equipment usedin many grinding processes. We study a 36 semi-autogenous mill used in primary grinding for mineralprocessing, a ball mill used for cement clinker grinding, a grinding table also used for cement grinding,a tower mill used for fine grinding both in mineral processing and for industrial minerals and finally foran Isamill, which is used for ultra-fine grinding in mineral processing. Copyright q 2008 John Wiley &Sons, Ltd.

    Received 22 March 2007; Revised 12 November 2007; Accepted 20 November 2007

    KEY WORDS: comminution; milling; particles; breakage; energy utilization; DEM

    1. INTRODUCTION

    Particle breakage is an essential process in many different industries, ranging from mineral

    processing, quarrying, cement to food processing and pharmaceuticals. The aim is to reduce

    the original feed stock particles from the sizes supplied to the much smaller sizes required for

    either further processing or to be a part of final products. In some industries, particularly mineral

    Correspondence to: P. W. Cleary, CSIRO Mathematical and Information Sciences, Private Bag 33, Clayton, Vic.3168, Australia.

    E-mail: [email protected]

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    320 P. W. CLEARY, M. D. SINNOTT AND R. D. MORRISON

    processing, the intent is to do this as efficiently as possible (that is, with respect to energy consump-

    tion and wear materials), producing optimal sizes for liberating target minerals at maximum mill

    throughput and minimizing operating costs. For a good introduction to the issues of comminution

    in mineral processing, see [1]. In industries such as food and pharmaceuticals, the mill sizes are

    much smaller and capital and energy costs are unimportant, but the final product specifications aretight and the focus is on quality and cleanliness/avoidance of contamination.

    Size reduction is typically accomplished in a series of stages whose design is highly dependent

    on the nature and size of the feed stock and the reduction ratio that is required. The first stage

    in mining and mineral processing is usually some form of blasting to liberate ore and waste

    from the deposit. A typical mineral processing grinding circuit is tailored to the properties of

    the particular mine ore. Commonly, this involves two or more classes of equipment starting with

    crushers, followed by semi-autogenous (SAG) mills and then sometimes ball mills. Occasion-

    ally, high-pressure grinding rolls or other novel devices are substituted. For cement industries,

    multi-stage ball mills, roller mills or grinding tables and high-pressure grinding rolls are often

    used, sometimes in series. For fine and ultra-fine grinding, particles will typically pass through a

    number of stages of grinding, finishing with tower mills or stirred mills such as the Netzsch or

    Isamill.

    Regardless of the industry and which of the possible outcomes is required, the size reduction

    process needs to be better understood and optimized for those applications. Traditionally, this has

    been done using extensive laboratory, pilot scale and field trials. These are expensive and time

    consuming and allow models to be constructed that are limited by the range of conditions tested

    and the complexities of the many different processes occurring within these mills. For a good

    introduction to data fitted modelling of a broad range of comminution equipment, see [1]. Tradi-

    tional experimental and modelling approaches are not well placed to create the required insights

    into comminution fundamentals or facilitate radical step changes in performance. Computational

    modelling, using the discrete element method (DEM), has been growing in capability over the past

    decade and is now able to provide significant insights into the operation of existing and future

    mills and can be used as a tool to aid equipment and process design and optimization.DEM has been used for modelling a wide range of industrial applications, particularly in milling.

    Early work on ball mills by Mishra and Rajamani [2, 3] has been followed by Cleary [4, 5] and

    Cleary and Sawley [6]. Axial transport and grate discharge were studied in [7]. Similarly, SAG

    mills were modelled by Rajamani and Mishra [8] and subsequently by others [912]. Experimental

    validation has been performed for a centrifugal mill [13] and for charge motion and shape in a

    laboratory scale model SAG mill [1416] and for grinding rates [17]. Other applications include

    vibrational segregation [1820], flows in rotating drums [1921], hopper flows by Ristow [19]

    Potapov and Campbell [22] and many more. A broad range of applications simulated using DEM

    by Cleary et al. are reviewed in [23].

    2. DISCRETE ELEMENT METHOD

    DEM is a simulation method that models particulate systems whose motions are dominated by

    collisions. This involves following the motion of every particle or object in the flow and modelling

    each collision between the particles and between the particles and their environment (stationary

    and moving objects such as mills and crushers). The discrete element algorithm has three main

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 321

    stages:

    1. A search grid is used to periodically build a near-neighbour interaction list that contains all

    the particle pairs and objectparticle pairs that are likely to experience collisions in the short

    term. Using only pairs in this list reduces the force calculation to an O(M) operation, whereM is the total number of particles. Industrial simulations with 2 million particles are now

    possible in reasonable times on current single-processor workstations.

    2. The forces on each pair of colliding particles and/or boundary objects are evaluated in their

    local reference frame using a suitable contact force model and then transformed into the

    simulation frame of reference.

    3. All the forces and torques on each particle and object are summed and the resulting equations

    of motion are integrated to give the resulting motion of these bodies. Time integration is

    performed using a second-order predictorcorrector scheme and typically uses between 15 and

    25 time steps to accurately integrate each collision. This leads to small time steps (typically

    104106 s depending on the controlling length and time scales for each application).

    2.1. Collision model

    The general DEM methodology and its variants are well established and are described in review

    articles [2426]. The implementation used here is described in more detail in Cleary [4, 20, 23].

    Briefly, the particles are allowed to overlap and the amount of overlap x , and normal vn and

    tangential vt relative velocities determine the collisional forces via a contact force law. There are

    a number of possible contact force models available in the literature that approximate collision

    dynamics to various extents. We use a linear spring-dashpot model that is shown diagrammatically

    in Figure 1. For more complex models, see [26, 27].

    The normal force

    Fn=knx+Cnvn (1)

    consists of a linear spring to provide the repulsive force and a dashpot to dissipate a proportion of

    the relative kinetic energy. The maximum overlap between particles is determined by the stiffness

    kn of the spring in the normal direction. Typically, average overlaps of 0.10.5% of the particle

    diameter are desirable, requiring spring constants of the order of 106108 N/m in three dimensions.

    Figure 1. Contact force law consisting of springs, dashpots and frictional sliders used for evaluating forcesbetween interacting particles and boundaries.

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    322 P. W. CLEARY, M. D. SINNOTT AND R. D. MORRISON

    The normal damping coefficient Cn is chosen to give the required coefficient of restitution

    (defined as the ratio of the post-collisional to pre-collisional normal component of the relative

    velocity), and is given in [20].

    The tangential force is given by

    Ft=min

    Fn,kt

    vt dt+Ctvt

    (2)

    where the vector force Ft and velocity vt are defined in the plane tangent to the surface at the

    contact point. The integral term represents an incremental spring that stores energy from the

    relative tangential motion and models the elastic tangential deformation of the contacting surfaces,

    while the dashpot dissipates energy from the tangential motion and models the tangential plastic

    deformation of the contact. Depending on the history of the contact, it is possible for the spring

    to be loading in one direction and simultaneously unloading in the orthogonal direction. The total

    tangential force Ft is limited by the Coulomb frictional limit Fn, at which point the surface

    contact shears and the particles begin to slide over each other.

    2.2. Geometry representation

    Industrial applications place heavy demands on the geometric capabilities of DEM codes, partic-

    ularly for three-dimensional geometries. In our DEM code, any two-dimensional cross section

    can be constructed using an appropriate number of line segments, circular segments or discs.

    This gives essentially unrestricted geometric capability. In three dimensions, boundary objects are

    constructed by either extruding two-dimensional cross sections or using triangular finite-element

    style surface meshes. Such meshes can be produced using any reasonable mesh generator from

    solid models generated in most CAD packages. This provides enormous flexibility in specifying

    three-dimensional environments with which the particles interact.

    2.3. Particle shape

    The particles are represented either as spheres or super-quadrics, which in a standard frame of

    reference are given by

    xa

    m+

    yb

    m+

    zc

    m=1 (3)

    The ratios of the semi-major axes b/a and c/a are the aspect ratios of the particles and the super-

    quadric power m determines the shape of the particle. For m=2 elliptical particles are obtained.

    If additionally a=b=c, then the particles are spherical. As m increases, the shape becomes

    progressively more cubic, with the corners becoming sharper and the particle more blocky. By

    m=10, the particle shape is essentially a cube (for aspect ratios 1). This is a very flexible classof shapes, which varies continuously (m is not restricted to integer values). This allows plausible

    shape distributions to be represented and the shape can be made to change dynamically during

    simulation (for example, with abrasion as measured by the shear energy dissipation on the particle).

    The computational cost of simulations using super-quadrics is currently between 2 and 5 times

    that of using spheres. This shape description was introduced to DEM in two dimensions in [28]

    and in three dimensions in [23].

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 323

    2.4. Model outputs

    The nature of the quantitative predictions that one wants to make, using the extensive state data

    available in DEM simulations, depends very much on the application and the questions one is

    seeking to answer. For industrial applications we typically calculate: Boundary stresses (for mechanical design and fatigue prediction, including coupling to FEM

    for structure response prediction).

    Wear rates and distributions (for estimating the lifespan of equipment and wear components).

    Accretion rates (for predicting accretion-induced blockage, changes in open area of screens).

    Collisional force distributions, collision frequencies, energy absorption spectra (for under-

    standing breakage and agglomeration).

    Power consumption and torques (for equipment design).

    Flow rates and flow statistics (for summarizing characteristics of complex flows).

    Sampling statistics (to assess the accuracy of various sampling processes).

    Mixing and segregation rates (to assess the progress of intended mixing and de-mixing

    processes and to understand the degree of segregation and its effects on other processes where

    segregation and/or mixing are not intended).

    Residence time distributions (to assess the range of times that particles are exposed to various

    environments within a process, for example in granulation and in comminution).

    Axial transport rates (to assess the axial flow along cylinders (rotating and stationary) that

    are commonly used).

    For more details on the simulation method and on the data analysis see Cleary [20, 23].

    3. TUMBLING MILLS

    Tumbling mills are large cylindrical vessels that rotate about their axes at reasonable fractions

    of the speed required to centrifuge the particulate material that is ground within. The grinding isproduced by a range of mechanisms including high-speed impact, abrasion, chipping and nipping

    of small particles between large ones. Grinding is often augmented by adding various types of

    grinding media. There are several specific variants of the tumbling mill, which are distinguished

    by their size, the size of material fed to the mill and the type of grinding media. These include:

    Autogenous grinding (AG) mills, which are fed a mixture of large and small rocks with the

    large rocks acting as grinding media.

    SAG mills are the largest tumbling mills used (up to 40 or 12.2 m diameter) and are used for

    primary grinding, mainly in mineral processing, using a coarse feed (typically up to 250 mm

    diameter rock) coming from some arrangement of crushers. There is insufficient coarse rock

    to produce large impacts so the charge is augmented by steel grinding balls. These can make

    up between 30 and 50% of the volume of the charge. However, neither large rocks nor ballsproduce very many large impacts as discussed later.

    Rod mills are mills where the grinding media consist of large rods (often almost the length of

    the mill). These mills typically take a relatively fine feed material of perhaps 20 mm topsize,

    (see [23] for an example of DEM modelling of a rod mill).

    Ball mills are typically used for secondary grinding in mineral processing, taking feed from a

    SAG mill or an equivalent device. It is typically smaller than the SAG mill (38 m diameter)

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    324 P. W. CLEARY, M. D. SINNOTT AND R. D. MORRISON

    and has higher ball loadings using smaller balls and with a finer feed. Ball mills are also

    commonly used in cement clinker grinding. These mills tend to be at the smaller end of the

    size range and have less aggressively designed lifters.

    In this paper, we present results for a large-scale model of a 36

    (10.86 m) diameter SAG millused in mineral processing and for a 4 m diameter (first compartment) ball mill used for cement

    grinding for coarse and fine feeds.

    3.1. SAG mill

    The SAG mill used in this modelling is a common design 36 or 10.86 m diameter SAG mill, which

    would typically be used for hard rock mineral processing. This mill rotates at 10 rpm (which is 78%

    of the critical speed at which the particles within the mill will centrifuge). Since collisions of the

    rock with the mill wear the mill surface, the mill is lined with a replaceable liner, typically made

    of high-grade collision- and abrasion-resistant steel. The liner serves a dual purpose of protecting

    the expensive mill shell and gearing from damage, and locking into the charge to lift it higher anddeliver more energy into the charge to increase the rate of grinding.

    Here, we use a traditional 72 row (typically twice the diameter of the mill measured in feet)

    configuration of symmetric, close packed, steep face angle lifters (7). The overall fill level of the

    charge is 30% of the mill volume. The ball loading is 10% (of the mill volume) or around 13

    of the

    charge (by volume). The ball size distribution is given in Table I. A typical rock size distribution

    for a hard rock ore is also used and this is also given in Table I. The top size of the balls used is

    125 mm and the top size of the rock is 200 mm. The rocks are correctly resolved down to 6 mm

    with smaller rocks neglected.

    Previously [14] we have established, by detailed comparison of various DEM models with

    experiments for a scale model SAG mill, that two-dimensional models have significant quantitative

    errors. Therefore, we need to use some form of three-dimensional model. Since the mill geometry

    is symmetric along its length, we are able to model the charge motion in a 0.5 m wide axialslice of mill. A key difference between a three-dimensional slice model and a two-dimensional

    model is that the particles are represented as spheres rather than as cylinders and lock together

    in three-dimensional particle structures that are stronger than the simpler structures possible in

    two dimensions. Hence, although the overall charge shape is invariant in the axial direction, the

    Table I. Rock and ball size distributions for a 1.2 million particle SAG mill model.

    Balls Rocks

    125 to +88 mm 30% 200 to +140 mm 8%88 to +63 mm 40% 140 to +100 mm 12%63 to +44 mm 20% 100 to +70 mm 15%44 to +31 mm 10% 70 to +50 mm 15%

    50 to +35 mm 15%35 to +25 mm 10%25 to +18 mm 10%18 to +12.5 mm 10%

    12.5 to +8.5 mm 4%8.5 to +6 mm 1%

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 325

    three-dimensional particle interactions are important to predicting the correct flow dynamics. A

    second key need for using a three-dimensional model is that the energy transfer mechanisms during

    collision depend on the masses of the interacting particles. In two dimensions, the cylindrical

    particle masses vary as the square of the radius, whereas in three dimensions they vary as the

    cube. Hence, to obtain meaningful collision energy spectra from which we make deductions aboutcomminution performance, it is essential to perform the simulations in three-dimensions. This

    0.5 m slice is sufficiently wide to properly represent particle motion in a long mill. This type of

    model is therefore suitable for predicting breakage, power consumption, belly-lifter wear and to

    understand the main flow patterns of the charge. If one were interested in axial transport then the

    full length of the mill needs to be modelled [7]. The current SAG configuration was previously

    modelled [10, 12] using a similar rock size distribution but with the rock size truncated at a much

    higher size of 25 mm. In this paper, we present simulation predictions using the more realistic

    6 mm cut-off for the rock size distribution. This involves simulating 1.2 million particles in this

    system.

    Figure 2 shows the charge distribution looking along the axis of the SAG mill. The charge

    surface has a very clear bi-linear shape, rising slowly from the toe to the middle of the charge and

    then steeply up to the shoulder. Substantial cataracting streams depart from the lifters above the

    shoulder and fall in well-defined coherent bands to impact on the liner above the toe at around

    the 3 Oclock position. There is a central slow-moving band within the charge midway between

    the free surface and the mill liner with strong shear on either side. The cataracting stream reaches

    around 7 m/s as it hits the liner above the toe of the charge.

    Figure 2. Slice model of a 36 SAG mill with 1.2 million particles coloured by speed with red being19 m/s and dark blue being stationary.

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    326 P. W. CLEARY, M. D. SINNOTT AND R. D. MORRISON

    Figure 3. Energy spectra for 36 SAG mill: (a) frequency distribution of different collision energies and(b) rate of energy dissipation at each collision energy.

    The power prediction for this 0.5 m axial slice of the mill is 796 kW. This is very close to the

    average power draw reported in Cleary [10] of 778 kW for the case with a 25 mm cut-off for the

    rock size. This suggests that the lower cut-off has little effect on the power draw prediction. For

    the full 6 m axial length of the mill we would therefore expect a power draw of 9.54 MW.

    Figure 3 shows the way in which the energy consumed by the mill is dissipated. The left plot

    shows the frequency distribution of the various collision energies. The peak energies, corresponding

    to the largest rocks and balls falling at the top of the cataracting stream, are 50 J. However, in

    comparison with the numbers of rocks, there are very few of these high-energy impacts. The most

    common collisions occur for very low energies around 0.1 mJ with frequencies exceeding 1 million

    collisions per second. There are enormous numbers of collisions recorded at even lower energies

    (down to the minimum of 1010 J). Of more interest is the energy dissipated at each of these

    energy levels. This is shown in the right plot and indicates that energy levels of around 100 W arebeing dissipated by the highest energy collisions. The strongest energy dissipation rate occurs at

    around 0.15 J with around 5 kW dissipated by these collisions. The energy dissipated per energy

    level decreases slowly until around 0.1 mJ and then drops linearly and more rapidly, so that by the

    106 J level only 1 W of energy is being dissipated, despite there being a frequency of a million

    collisions per second. Hence, although there are enormous numbers of collisions at these small

    energy levels, the total energy dissipated by these very gentle collisions is minimal.

    Comparing the distributions of shear and normal collision energies in Figure 3, it is clear that

    the shear energy levels are consistently higher than the normal ones. The shear energy dissipation

    rate (shown in the lower plot of Figure 3) occurs at 0.2 J, which is double that of the overall

    maximum dissipation rate and 20 times higher than the energy level for the maximum normal energy

    dissipation rate (at 0.01 J). The area between the shear and normal energy curves is the difference

    in the total energy consumption for impact and abrasion collision modes. Remembering that thevertical scale is logarithmic, the difference between these curves is significant. This demonstrates

    that a large majority of energy dissipation in a SAG mill occurs by particles sliding over and past

    each other and not by direct head-on impact of the particles.

    An alternative approach to the spectra is to consider energy levels by tracking some selected

    particles in each size class. This approach has demonstrated that breakage in a single impact

    is highly unlikely for most particles larger than a few millimeters [12]. Hence, a detailed

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 327

    Table II. Energy usage within the SAG mill.

    Collision type % Total energy in this type

    Rockrock 51.9

    Rockliner 1.2Rockmedia 38.3Mediamedia 8.3Medialiner 0.3

    investigation into incremental breakage was carried out. This work found a reasonably well-defined

    minimum energy below which essentially no impact damage occurs. For impacts with larger

    energy, the particle accumulates incremental damage, increasing the probability of breakage with

    each event [29].

    Table II shows the distribution of energy dissipation by the different types of collisions occurring

    in the mill. Around 52% of the energy is dissipated in rockrock collisions with another 38%

    dissipated in rockmedia collisions (leading to both rock comminution and ball wear). A further8.3% of the energy is dissipated in ballball collisions with some fraction of this leading to

    ball wear. Only 1.2% of the energy is lost in rockliner collisions, with some fraction of this

    contributing to liner wear. A further 0.3% of the energy dissipated is in medialiner collisions and

    contributes to ball and liner wear. DEM can be used to explore modifications to liner geometry,

    mill operating parameters, feed and ball size distributions to help move energy dissipation from

    non-useful categories (such as liner collisions) to rockrock and rockmedia collisions, which

    should improve grinding rates for the mill.

    3.2. Ball mills for cement grinding

    Ball mills are often used for grinding clinker as part of the cement production process. Often

    two chamber mills are used with the chambers separated by a grate. Typically, finer ball chargesare used in the second compartment to give a finer grind than is possible in the coarser first

    compartment. Here we simulate the first compartment configuration of such a ball mill. The liners

    used for cement mills tend to be less aggressive than those used in mineral processing. Here we

    use a common wave liner in a mill with an internal shell diameter of 4.0 m. The liner consists of

    36 liner plates, each with a plate length of 340 mm and a height of 65 mm, with a single wave per

    plate. The overall fill level used is 35% (of the volume of the mill) with the charge consisting of

    half balls and half clinker. The ball size distribution is uniformly mass weighted with a smallest

    size 50 mm (2 in) and a top size of 95 mm (3 34

    in). Two different clinker sizes are used:

    a coarse feed with particles in the range 2580 mm and

    a fine feed with particles in the range 1050 mm.

    The fine feed case is more typical of cement clinker grinding.A periodic slice of the mill is again sufficient to predict the dynamics of interest. The periodic

    slice length used was 250 mm. The mill rotation rate was 16.28 rpm, which is 76% of critical speed

    (based on a diameter of 3.9 m corresponding to the lowest point in the wave liner). For the coarse

    feed, the spring stiffness used was 1.5107 N/m and for the fine feed a value of 2.0107 N/m was

    required to give the desired average particle overlaps of 0.50.7%. The friction coefficient was 0.5

    for all materials. The coefficient of restitution was chosen to be 0.8 for steelsteel collisions, 0.5

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    328 P. W. CLEARY, M. D. SINNOTT AND R. D. MORRISON

    for steelclinker and 0.3 for clinkerclinker collisions. The number of particles was 11 200 for the

    coarse charge and 75 000 for the fine charge. The speed of a DEM code can be quantified using

    the Cundall number (Cu), which is the number of particle time steps per cpu second. On a single

    processor 3 GHz Pentium, these ball mill simulations produce Cundall numbers of 450 000.

    Figure 4 shows the charge distribution in the ball mill for the coarse and fine size feeds. Thecharge shape is very similar in both cases with the familiar bi-linear shape for the free surface

    rising from the toe to the inflexion point near the centre of the mill and then steeply up to the

    shoulder position. Despite the difference in mill size and the size differences in the charge, this

    flow pattern is fairly similar to that of the SAG mill, but with the inflexion point now closer to

    the mill axis and the angular difference between the two surfaces of the charge being less severe.

    It is mainly in the energies delivered to the particles that the mills behaviour differs. For both

    Figure 4. Charge distribution in the ball mill with (left) coarse size distribution and (right) fine sizedistribution. The top row is coloured by the particle speed (with red being 7 m/s and dark blue being

    stationary) and the bottom row is coloured by particle class (with dark blue showing the balls).

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 329

    coarse and fine feeds, the shoulder position lies in the range of 210220, as measured clockwise

    from the 3 Oclock position. This range reflects the difference in the timing of the disengagement

    of the charge from different points on the wave lifter. For the high point of the lifter, which is

    closer to the centre of the mill, the disengagement occurs a little earlier at around 210 , but for the

    more distant low point of the line, the disengagement occurs around 220. The toe position for thecoarse feed lies in the range of35 to 40, whereas for the fine feed the toe lies consistently in

    the tighter range of35 to 36. The finer feed is more able to cushion the large balls and this

    damps fluctuations in the avalanching flow, leading to lower variability of the stopping position

    at the toe. The other visible effect of the finer feed is that the modest cataracting stream is more

    clearly defined because of the much larger numbers of much smaller particles in this region. The

    peak particle speeds are consistently around 6.57.0 m/s at the bottom of the cataracting stream.

    The pictures in the lower row of Figure 4 show the particles coloured by their type. The dark blue

    particles are the balls and they are fairly uniformly distributed throughout the charge. With such

    a large ball load, it is difficult for the balls to concentrate around the centre of recirculation due

    to size segregation driven by the centrifugal force (as sometimes occurs for SAG mills).

    The power draw predicted for the coarse feed is 146.9 kW/m of mill length or 1.028 MW for a

    7 m long ball mill. This corresponds to a torque of 86.2 kN/m average torque (averaged over 6 s of

    operation of the mill after steady state has been achieved). For the fine feed, the power draw rises

    slightly to 148.2 kW/m of mill length, corresponding to an average torque of 87.0 kN/m. This

    represents a 0.9% increase when moving to the finer feed. We attribute this very minor change

    to the fact that the finer particles are slightly better able to pack the voids between the balls and

    this will lead to a slightly higher bulk density for the charge whose centre of mass will therefore

    move slightly closer to the mill shell, thereby increasing the mill power draw. This is a real but

    weak effect and will continue as the feed is made finer or finer product particles are included in

    the simulation.

    Figure 5 shows the selected energy spectra for the different types of collisions for both feed

    size distributions. The energy dissipated in every collision is tracked and the frequency distribution

    of the different collision energy intensities is calculated. This is done for the normal work (as ameasure of impact breakage), shear work (as a measure of abrasion/attrition damage) and total

    work (measuring overall breakage). A variant of the spectra that we use here involves multiplying

    the energy of collisions by the frequency to give the energy dissipation rate for each collision

    energy level in the mill. This helps us understand where the 147 kW of power is being consumed

    within the mill. Note that the energy dissipation rates quoted here are for just the particles in a

    single periodic slice. For a full mill, these need to be scaled by the ratio of the mill to slice lengths.

    To start, we concentrate on the coarse feed case (left column of Figure 5). All the energy

    spectra have a log-normal-like distribution. There is a maximum energy that any particular type

    of collision can dissipate, typically governed by the maximum fall height of the heaviest particle

    involved in that type of collision. These events are very rare with frequencies typically around 0.1

    per unit length of the mill. This is the equivalent of one per 10 s or once per two to three mill

    revolutions. Although rare, these events are fairly strong so the total energy dissipated by these isaround 0.11.0 W. For lower energy levels the frequency of collision rises rapidly to a peak level

    of around 105 collisions/s/m at 7 mJ total collision energy. The highest rate of energy dissipation

    of 200 W occurs at a higher energy level of 0.17 J. From these median peaks, the collision rates

    decline slowly at first and then more rapidly. The combination of declining collision energies and

    frequencies means that the energy dissipation drops off rapidly from the peak. Figure 5 shows the

    spectra for the four most important types of collision in the mill. The strongest energy dissipation

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    Figure 5. Energy spectra for (left) coarse size distribution and (right) fine sizedistribution in the cement ball mill.

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 331

    rate is experienced for the rockmedia collision at around 150 W occurring at an energy level

    of 0.2 J. The next most important energy dissipation is in the rockrock collisions whose peak

    energy dissipation is 100 W occurring at 0.1 J. The next most important contribution occurs for

    mediamedia collisions with peak energy dissipation of 70 W at a collision energy level of 0.36 J.

    This energy contributes only to ball wear and results from a much smaller frequency of much moreintense impacts (due to the large weight of the balls). Rockliner collisions are more intense with

    their peak at 0.4 J but these are much less common so their contribution to the overall dissipation

    is only 6 W.

    As was observed earlier for the SAG mill, the energy absorption is again dominated by the

    shear component. For all the collision types, the distribution of shear energy dissipation is very

    similar to the overall energy dissipation distribution with the peak shear dissipation rate occurring

    at between 1.0 and 2.0 times the overall level. Conversely, the normal energy dissipation is much

    smaller with its peak rate of dissipation being around 40% of that of the shear and occurring at

    collision energy levels around 10 times smaller. This again demonstrates that the energy dissipation

    is predominantly produced by particles sliding over each other rather than during head-on impacts.

    Sliding motion of the balls is the dominant mechanism leading to size reduction in the ball mill.

    Comparing the energy spectra for the fine and coarse feeds (Figure 5), we observe a range of

    changes. The maximum energy found for rockrock collisions declines by a factor of 2, which

    is smaller than one might expect since the mass of the largest rocks has decreased by a factor of

    4 and that of the lightest rock by a factor of 10. This indicates that the presence of the media is

    having a significant effect on the rockrock collisions by increasing the pressures involved. The

    highest energy dissipation rate is observed to move to a lower energy (from 0.1 to 0.01 J), which is

    about a quarter of what might be expected on the basis of the reduction in the product of the rock

    masses. The actual peak energy dissipation has increased slightly indicating that the distribution

    of energy dissipating collisions has become tighter.

    The maximum energies for rockmedia collisions have dropped by a factor of 2 (similar to the

    change for rockrock) but the highest energy dissipation collision level has only declined by a

    factor of 3, which is again around half the level that one might expect based on the rock masschange. The peak energy dissipation rate is unchanged in magnitude but the location has moved

    to lower-energy levels.

    The mediamedia spectra are broadly unchanged, with a modest reduction in the energy dissi-

    pation in the low energy tail and a slight narrowing of the spectra. This demonstrates that the ball

    component dominates the charge behaviour and has large impacts on the rock behaviour but the

    rock changes have very limited effects on the media.

    For the rockliner collisions, the maximum shear energies have declined by about one-third,

    whereas the normal energy has declined by a factor of 3. This indicates that direct liner impacts by

    larger rocks in the toe region have been substantially reduced in line with the reduction in the rock

    masses. The shear energy dissipation which is controlled by the pressure of the charge above the

    rock particles sliding along the liner only varies slightly since the ball mass is largely unchanged.

    Table III shows the distribution of energy consumption by the different types of collisions inthe ball mill for the two feed sizes. These values are the integrated areas under the energy spectra

    curves. For the coarse feed about one-third of the energy is dissipated in rockrock collisions, with

    a large 46% dissipated in rockmedia collisions. The mediamedia dissipation is also significant at

    17.8%. The rockliner and medialiner energy dissipations (leading to liner wear) are very small.

    For the finer feed, the fraction of energy dissipated in rockrock collisions has jumped sharply to

    51% with matching falls in the rockmedia and mediamedia dissipation of 8 and 9%, respectively.

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    Table III. Energy usage within a cement ball mill.

    Collision type Coarse feed (%) Fine feed (%)

    Rockrock 33.5 51.4

    Rockliner 1.9 1.6Rockmedia 46.2 38.2Mediamedia 17.8 8.5Medialiner 0.5 0.2

    Table IV. Distribution of energy absorption among different particle classes and the mass and areaefficiencies of energy absorption for the ball mill with the coarse feed.

    Class Size range (mm) % Total energy % Mass % Energy/% mass % Area % Energy/% area

    1 +50 to 95 balls 41.21 74.30 0.555 37.27 1.106

    2 +60 to 80 feed 8.44 5.15 1.639 7.84 1.0773 +40 to 60 feed 22.82 10.27 2.221 21.66 1.0544 +25 to 40 feed 26.33 10.28 2.562 33.24 0.792

    Table V. Distribution of energy absorption among different particle classes and the mass and area efficienciesof energy absorption for the ball mill with the fine feed.

    Class Size range (mm) % Total energy % Mass % Energy/% mass % Area % Energy/% area

    1 +50 to 95 balls 27.74 74.29 0.373 24.24 1.1442 +40 to 50 feed 7.86 3.86 2.038 5.99 1.313

    3 +30 to 40 feed 15.86 6.43 2.468 12.90 1.2304 +20 to 30 feed 22.98 7.71 2.980 21.59 1.0645 +10 to 20 feed 24.65 7.71 3.196 35.28 0.699

    This would suggest that the media used here is much better at supplying energy to the fine feed

    material than to the coarse and also suggests that the coarse feed will produce higher ball wear

    rates. The liner energy dissipation components have also dropped by 25%, which would suggest

    that the coarser feed will produce higher wear rates on the liner and a shorter lifespan for the liner.

    This comparison demonstrates that although the overall power draw for the two feeds are almost

    the same, the nature of the collisional environment in the charge and how this energy is dissipated

    is substantially different with the finer feed being better able to absorb energy with perhaps 56%absorption leading to better grinding than for the coarse feed with energy absorption of only 42%.

    On this fairly crude energy calculation one might anticipate a one-third increase in grinding for

    the fine feed (which should result in some combination of higher throughput and a finer grind).

    To help understand the energy absorption by the rock, we look now at the energy dissipation in

    each of the size classes of the charge. Tables IV and V show the distribution of energy absorption

    for the different particle classes and the mass and area efficiencies of energy absorption for the

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 333

    coarse and fine feeds, respectively. The actual fraction of the energy dissipated is heavily dependent

    on the amount of that material in the mill. We therefore look at two measures of the energy

    absorption efficiency of each of the charge components, one based on energy absorption per unit

    mass and the other per unit area. From Table IV we see that the balls absorb a much lower fraction

    of energy per unit mass than average (around half). The coarsest fraction of rock absorbs around1.6 times the average. This continues to increase to 2.56 for the smallest of the rock classes.

    Hence, the energy is dissipated disproportionately to the mass with the finer sizes absorbing more.

    There are a number of contributors to this including the difference in density of the rock to the

    steel, the different coefficients of restitution for collisions of different combinations of material

    and the different numbers of contacts that each particle class has with neighbours. Looking at the

    energy absorption per unit of surface area we find a much better correlation. The ball and first

    two rock classes have area efficiencies of a little over 1, meaning that they absorb energy broadly

    in proportion to the surface area of each class. This behaviour was confirmed in more detail in

    small-scale autogenous mill testing reported by Djordjevic et al. [30]. Relating the particle wear

    rate to the DEM estimate of frictional energy works quite well in these energy ranges.

    The smallest rock fraction though has a much lower efficiency of around 0.8. This probably

    relates to the absence of smaller rock classes that make it harder to load the surface of this smallest

    class.

    Table V shows the energy absorption for the fine feed. The trends are similar to that observed

    for the coarse feed. The mass efficiency for the balls has dropped to only 0.37 while the largest

    rock class has a mass efficiency of 2.0, which is similar to that of the matching size class in

    the coarse feed. The mass efficiency continues to rise with decreasing size to a peak of over 3

    for the 1020 mm class. The correlation with area is again much better with the ball efficiency

    being around the same level of 1.1 as was found for the coarse feed. The largest rock class has an

    area efficiency of 1.3, meaning it is 30% better at absorbing energy per unit of surface area than

    average. The area efficiency declines very slowly with decreasing size (as it did for the coarse

    feed). The smallest size class has an efficiency of only 0.7, which is fairly similar to the behaviour

    of the smallest class in the coarse feed. This further suggests that this might relate to the absenceof smaller size classes.

    4. GRINDING TABLE

    Another grinding device, commonly used for cement grinding, is a grinding table or sometimes

    called a roller mill. In this particular example, a 4 m diameter table rotates around its centre at

    20 rpm. It has a lip around the outside to prevent particles being thrown off. Four 1.2 m diameter

    rollers are positioned just above the table. They counter-rotate so that the peripheral roller speed

    matches the table speed at the lowest point. Particles (with diameters 2575 mm and density

    3000kg/m3) are fed onto the centre of the table at a rate of 450 tph. Upon contacting the table, they

    are thrown outward. The lip on the edge of the table traps the particles, leading to the formationof a bed of particles on the table. Many of the outward flowing particles enter the region in front

    of the rolls, are drawn in by the motion induced by the roller rotation and are trapped between

    the table and the rollers where they are crushed due to compression. In this case, we impose

    a lower limit on the fragment particle size of 5 mm to ensure that the particle numbers remain

    computationally reasonable. The largest particles are typically the weakest and have a breakage

    force limit of 15 kN rising to 60 kN for the finer particles that have less remaining flaws.

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    Figure 6. Grinding table rotating at 20 rpm with four equi-spaced 1200 mm rollers and a central feedshown obliquely from above with particles coloured/shaded by speed.

    Figure 6 shows the flow (where the particles are coloured by speed) in such a mill once the bed

    depth has stabilized and the system is in approximate steady state. The incoming speed of the central

    feed stream is relatively fast. The particles slow as they strike the accumulated particle bed on the

    surface of the grinding table. As they are pushed outward, they accelerate until they encounter the

    rolls, are trapped and are dragged under the rolls. Here they are crushed by compression. Some

    high-speed fragments fly out from the rollers. Each roll generates a bow wave effect with a mass

    of particles building up higher in front of the roll than is found throughout the rest of the bed.

    Some of these particles are pushed sideways and flow around the roll forming the sides of the

    bow wave and some are pushed in front of the roll. The pile of material pushed along in front

    of the roll needs to generate enough back pressure in order to force particles under the roll. For

    this four-roll case, the bow wave from each roll is directed under the middle of the following roll.This would appear advantageous for particle breakage.

    The DEM model used here includes a simple but effective dynamic breakage model [10] where

    particles reaching a critical breakage force (as determined by the DEM contact force model) are

    replaced by a suitable assembly of daughter particles that are then free to move independently. In

    this case of spherical particles fracturing into spherical progeny, a geometric packing algorithm is

    employed. A primary fragment is chosen and placed in the parent. The next biggest particle that

    can fit is added along the same axis as the first. Then layers of maximally sized finer particles

    are added into the biggest gaps in the previous layer. This is performed repeatedly with up to

    five levels of progressively finer fragments added. For more details on the progeny construction

    method, see [10].

    For the grinding table, this means that the daughter fragment particles can then be re-trapped

    further under the roll and be broken again as many times as is needed for them to pass through thegap between the table and the rolls. Figure 7(a) shows the bed of particles on the table from below

    with the particles coloured by size (red for the largest and dark blue for the smallest fragments).

    The central feed and the outward flow of this coarser material (light blue to red particles) are

    visible in the middle sections of the table. The bow wave of coarser bed material piled up in front

    of and flowing around each of the rolls is also clearly visible. The compression region under the

    roll can also be observed with the larger particles able to approach only within several diameters

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 335

    Figure 7. Grinding table with particles coloured/shaded by size shown from: (a) underneath and(b) looking through the roll at the bow wave/zone of compression breakage under the roller.

    of the middle of the roll before the rolltable distance becomes smaller than the particles size

    and it is fractured. There is a clear gradation of size as the particles move further under the

    rolls. Finally, only a layer of dark blue product particles is observed as the particles reach the

    minimum size enabled for this model. These fine fragments then pass under the roll and either

    join the concentrated band of fines trapped by the lip around the edge of the table or flow over

    this packed bed of fines and are discharged from the table as product. The discharge material

    commonly includes both unbroken feed and product and then needs to be separated by some typeof air classification system with the coarse fraction returned to the feed stream in a closed circuit.

    At this stage, we do not seek to model either the ultra-fines or the coupled air flow.

    Figure 7(b) shows the breakage region and bow wave in front of a roller from a view looking

    through the transparent roll with the particles again coloured by size. The restriction of the flow

    under the roller causes a bow wave of densely packed particles to form in advance of the roller.

    Many of the particles in the bed are pushed sideways around the roll, some being forced over

    the lip of the table to be discharged and some radially inwards onto the table. Some particles are

    dragged under the roll and rapidly crushed since the gap between the roll and the table decreases

    as the particle is dragged further under the roll. The fines (coloured dark blue) are free to flow out

    from the back of the roll.

    Figure 8 shows the energy spectra predicted for the grinding table. The normal energy dissipation

    (typically associated with impact breakage) is shown in black and the shear energy absorptionis shown in grey. It is clear that the shear energy is weighted to higher levels than the normal

    energy. Measured over all collisions, the normal impact energy accounts for 35% of total energy

    consumption while the shear dissipation accounts for 65%. The shape of the energy spectra is

    noticeably different to those of the ball mill (see Figure 5) even when considering the different

    minimum energy thresholds used in the plotting. This indicates that there are important performance

    differences between the two types of mill even though they are often used for the same cement

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    Figure 8. Energy spectra for the grinding table: (a) over all collisions; (b) from rock-grinding table androlls; and (c) from rockrock collisions.

    applications. The energy spectra for the collisions between the particles and the grinding surfaces(tables and rolls) are very similar to the spectra for all collisions. The spectra for the particle

    particle collisions though are quite different. The collision rates are much lower as are the average

    levels of energy dissipation (around 10 times lower) but there is also a long tail of high-energy

    events. These occur between particles in the compression zone as they are loaded under the rolls.

    The rate of these collisions is low but they are an important part of the breakage and energy

    consumption process.

    Note that the smallest particle size can become jammed under the rolls in an unphysical manner

    because of its inability to break and so can absorb unrealistic amounts of energy. Hence, ideally the

    minimum particle size used in the model should be less than the separation of the rolls from the

    table (i.e. less than the bed depth under the rolls). This will allow more accurate prediction of

    the energy absorption by the smallest class and therefore more accurate predictions of the power

    draw.Figure 9 shows the distribution of energy absorption across all the breakable size classes. The

    largest particle sizes absorb only around 5% of the energy (despite being around 10% of the particle

    mass). The rate of absorption rises almost linearly for particle classes down to 40 mm diameter.

    At this size, the particles absorb around 10% of the energy and represent 10% of the particle

    mass. For smaller sizes, the energy absorption rises sharply, with 27% of the energy absorbed

    for the smallest of these size classes. The particle sizes used in this grinding mill simulation are

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    0.0

    5.0

    10.0

    15.0

    20.0

    25.0

    30.0

    30.0 40.0 50.0 60.0 70.0

    Particle class diameter (mm)

    Energyabsorption%

    Figure 9. Distribution of energy absorption across all the breakable size classes.

    very similar to the ones used in the coarse cement ball mill discussed earlier. The distribution of

    energy absorption with particle size is remarkably similar for the two machines (see Table IV for

    the ball mill results) with the % energy/% mass being around 0.5 for the largest particle sizes

    and increasing to around 2.5 for the finest. This suggests that the rate of energy absorption for the

    grinding table is again controlled by the total surface area of each size class.

    Figure 10 shows the rate of fracture of the particles and the rate of product creation by the

    mill. The overall rate of fracture is quite variable with high-frequency fluctuations, depending

    sensitively on the specific particles being trapped under the rolls. The overall rate varies between

    250 and 450 fracture events per second. The fracture rates are smallest for the largest particles

    (since there are relatively few of these per unit mass). Figure 10(c) shows the fracture rate for

    the largest size with an average rate of 2 fractures/s. The breakage rate increases with decreasingsize. For class 7 (5562 mm) the rate has increased to 5 fractures/s. For class 4 (3036 mm) the

    much larger numbers of particles available mean that breakage is more continuous, albeit still with

    wide fluctuations and with an average of around 50 fractures/s. For the second smallest class, the

    fracture rate has risen to 130 fractures/s. The fracture rates are clearly dependent on the number

    of particles in each size class and also on their breakage strength. DEM is able to predict not

    only the energy consumption, the pattern of usage of that energy, but also the breakage outcomes.

    This work demonstrates that DEM models can, in principle, make predictions of the resident

    size distributions and product size distributions of mills. Determining how realistic these predic-

    tions are and refining the dynamic breakage models to improve them are the subject of ongoing

    work.

    For the current grinding table configuration, with its relatively light bed loading and for the mate-

    rial hardnesses chosen, the product generation rate predicted is about 11 kg/s or about 40 tonnes/h.The power draw measured was 65 kW with 25 kW being supplied by the grinding table and 10 kW

    supplied by each of the four rolls. This size of table can be run with much higher bed loads so as to

    generate product at rates up to 400 tonnes/h. Scaling up the power similarly would lead us to expect

    a power draw of around 650 kW. This is around 6 times lower than the typically installed power.

    This leads us to expect that truncating the product size distribution may lead to under-prediction

    of power draw for these types of compressive mills since much of the finer breakage occurring in

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    Figure 10. Particle breakage predicted for the grinding table: (a) overall rate of fracture eventsfor all particles; (b) the rate of product generation (given by particles in the smallest class);(c) biggest class of particles (class 10); (d) class 7; (e) class 4; and (f) class 2 which are the

    smallest particles allowed to fracture by the model.

    the real mill is not well represented in the model with this choice of minimum particle size. This

    suggests that quantitatively accurate prediction of all mill outputs may require resolving more of

    the finer end of the size distribution. This is becoming increasingly feasible as computer power

    increases. More investigation is needed to clarify these issues.

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 339

    5. STIRRED MEDIA MILLS

    Stirred mills are becoming increasingly used for fine and ultra-fine grinding. This technology is still

    poorly understood when used in the mineral processing context. This makes process optimization

    of such devices problematic. Three-dimensional DEM can again play a part in understanding thefundamentals of particle size reduction and media behaviour inside these mills. In this section, we

    present examples of DEM simulation for two fine-grinding mills.

    5.1. Tower mill

    A tower mill has a vertical cylindrical shell and employs a vertical double helical steel screw

    agitator located along the axis of the mill shell. The mill is substantially filled with grinding

    media (typically steel or ceramic balls). The screw agitator stirs the media while simultaneously

    lifting and circulating it throughout the mill. Fine feed material is introduced at the top surface of

    the charge and is circulated throughout the mill where it is ground by being crushed or sheared

    between media particles. The rotation of the screw creates centrifugal forces that compress the

    charge radially outwards. This is balanced by gravity. The screw rotational speed is limited by

    the need for the vertical force resulting from the centrifugal compression to remain smaller than

    gravity so that the charge is not lifted out of the mill, (see [31] for more details). The rotational

    speeds and therefore the energy intensity produced within the tower mill is much higher than for

    a ball mill (with a speed differential here of around a factor 6) and so a tower mill can typically

    produce a much finer product. The nature of the flow pattern of the media and manner of energy

    dissipation within the mill though is not well understood.

    Here we model a tower mill filled with dry ceramic media. A coefficient of restitution for the

    media of 0.3 is used (to take account of the fine powder being ground that will occupy the interstices

    between the balls in the mill). The friction coefficient was chosen to be 0.5 for mediamedia and

    medialiner collisions. The fine feed material which is intended to be ground by the mill is too

    small to be able to be included directly in the DEM model and is omitted. The grinding media,which are typically 12 orders of magnitude larger, can be fully included. A spring constant of

    10000N/m was used giving average overlaps of about 1%. Key geometric details and operating

    parameters for the tower mill are given in Table VI. The details of the ball charge are given in

    Table VII. The mill is filled nearly to the top and contains around 30 000 media particles.

    Figure 11 shows the distribution of the media in the tower mill once it has reached the steady

    state. The mill and charge have been cut away to allow us to see inside. Figure 11(a) shows the

    media coloured by speed. There is a solid core of media, occupying a large fraction of the mill

    vessel, which moves with the screw agitator and which is coloured green. Near the outer edge of

    the screw (which is the most rapidly moving part of the screw and the part that encounters the

    highest collision intensity) the media particles are driven faster, with particles achieving double the

    average core speed (and coloured yellow to red). Around the central moving core is a fairly clearly

    defined layer of slow-moving particles that are adjacent to the cylindrical mill shell. Frictionaldrag slows these particles. The sharp change in speed from the slow outer layer to the inner core

    means that there is significant shear in the narrow transition region between the blue and the green

    particles, which will lead to significant abrasive wear.

    Figure 11(b) shows the same particle distribution but now coloured by the amount of energy that

    each particle has absorbed from the normal component (head-on impact) of the collisions that they

    have experienced. There is a clear gradient in the energy absorption with the highest levels (which

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 341

    will correspond to regions of high-impact damage for the fine feed material that occupies the space

    between the balls in the real mill), with the highest levels found at the top of the mill and the lowest

    levels at the bottom. This occurs because the screw agitator is actually a very effective transport

    device and lifts the media in the core of the mill upwards, producing higher pressures at the top

    of the mill and therefore more intensive impacts. Figure 11(b) shows the distribution of the shearenergy absorption in the mill, which will correspond to the intensity of abrasive size reduction

    of the feed material. There is still a vertical gradient present with the highest values found at the

    top of the mill, but it is much weaker than for the impact (normal) energy absorption. The reason

    for this is that the Coulomb limit on the frictional force is proportional to the normal load. The

    shear energy absorption is a product of the shear force and the shear velocity. The vertical gradient

    in the normal force therefore creates a matching gradient in the shear component. However the

    shear velocity also contributes. Looking at the speed distribution (Figure 11(c)) we see that the

    transition from slow-moving blue material around the shell to fast-moving core material is much

    sharper at the bottom of the mill than at the top. Therefore, the shear rate is higher at the bottom

    than at the top. This increases the shear energy dissipation at the bottom of the mill. The mild

    vertical gradient in the shear energy absorption is therefore a balance between the normal force,

    which increases with distance from the bottom and the shear rate which decreases.

    One way of characterizing the flow in this type of mill is to calculate averaged one-dimensional

    profiles of the velocity components, as a function of mill height and radius. These are shown in

    Figure 12. The velocities are collected in radial and vertical slices of the mill to calculate a mean

    velocity and standard deviation for each radius and height. The mean radial velocity is statistically

    zero over most of the mill height. Near the base of the mill, it has a small but meaningful negative

    value, indicating that the particles are flowing from near the outer mill chamber radially inwards

    towards the central screw. Near the free surface, the radial velocity becomes significantly positive

    as particles flow back from around the screw, outwards towards the outer mill shell. The standard

    deviation is a measure of the variability of the velocity from its means and of the extent of

    fluctuations in the flow. The standard deviation is much larger than the peak values of the radial

    velocity and is constant with height. The bounding envelope, defined by the limits of the verticallines, indicates a region of phase space which is symmetric around zero that the particle radial

    velocities are likely to occupy. There is no meaningful mean radial velocity at any radius. This

    means that in any vertical column of particles there is equal radial movement in both positive and

    negative directions. The outside of the screw makes the largest contribution to driving the flow

    and also drives the fluctuations in the flow.

    The mean tangential or swirling velocity is low at the base of the mill and increases steadily

    over the bottom quarter of the mill. The swirling motion is then essentially constant over most of

    the height of the screw. There is a strong drop in the particle swirl in the top surface layers of the

    charge where the particles are more mobile and move out towards the mill shell. The variability of

    the swirl speed is also quite constant over most of the mill height, except in the surface layers. The

    tangential speed variation with radius is very strong. It is zero for radii less than or equal to 0.02 m

    since this is the radius of the core of the screw and particles cannot be found at these positions. Theswirl is low near the core of the screw (r>0.03) and increases strongly and monotonically with

    mill radius up to a peak at the edge of the screw (r=0.07). It then decreases almost linearly with

    radial distance out towards the mill chamber wall. There is a reasonable degree of slip experienced

    between the charge and the wall at about a third of the peak tangential speed.

    The mean axial velocity is statistically zero at each height and the amount of variability in

    the instantaneous axial velocities is also constant with height. Since the screw and chamber

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    342 P. W. CLEARY, M. D. SINNOTT AND R. D. MORRISON

    -0.1

    -0.05

    0

    0.05

    0.1

    0 0.5 1

    Height (m)

    Vr(m/s)

    -0.1

    -0.05

    0

    0.05

    0.1

    0 0.05 0.1 0.15

    Mill Radius (m)

    Vr(m/s)

    -0.4

    -0.2

    0

    0.2

    0 0.5 1

    Height (m)

    Vtang(m/s)

    -0.5

    -0.4

    -0.3

    -0.2

    -0.1

    0

    0 0.05 0.1 0.15

    Mill Radius (m)

    Vtang(m/s)

    -0.2

    -0.1

    0

    0.1

    0.2

    0 0.5 1

    Height (m)

    Vz(m/s)

    -0.2

    -0.1

    0

    0.1

    0.2

    0 0.05 0.1 0.15

    Mill Radius (m)

    Vz(m/s)

    Figure 12. One-dimensional profiles of media velocity (Vr, Vtang, Vz) versus mill height and radius forthe tower mill at t=10s. Confidence limits of one standard deviation show the variance in velocity. The

    radius of the screw is 0.07 m and the chamber radius is 0.12 m.

    cross sections do not vary with height, this means that in any axial slice there is, on average,

    as much material moving down the mill as there is moving up the mill with the screw motion.

    In contrast, the mean axial velocity varies strongly with radius. It has intermediate values

    within the screw, which increase moderately with radius to a peak transport rate at the outer

    edge of the screw. The mean axial velocity then drops almost linearly to a point just inside

    the mill shell. From a radius of around 0.09 m, the mean axial speed is negative, indicatingthat there is strong coherent down flow in an outer annular region that is about 25% of the

    radius of the mill chamber. The average speed of the outer down flow is much slower than

    that of the inner up flow because the mass flow rates must be equal (since this flow is in

    equilibrium) and the outer annulus has a much larger large cross-sectional area. The down-

    ward mass flux through this outer cylinder therefore matches the upward flux in the inner

    core.

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 343

    Combining the interpretations of the radial and axial mean velocity fields, it is clear that there is

    a strong global recirculation pattern within the tower mill driven by the screw motion. Particles are

    transported upwards by the screw in a central cylindrical core. As they approach the free surface,

    they begin to flow outward towards the mill shell. Once they are further out than 75% of the mill

    radius, they begin to flow somewhat erratically downwards in an annular down flow region aroundthe outside of the charge. At the bottom of the mill, the particles flow back in towards the screw

    where they are again picked up and transported upwards by the screw.

    To understand the nature of the energy utilization in the tower mill, we calculate one-dimensional

    profiles of the energy absorption rate as functions of mill height and radius. The distribution of

    energy absorption by the media indicates where in the mill the particle size reduction of feed

    material is occurring. Figure 13 shows that mean energy absorption rates for normal, shear and total

    energies are largely insensitive to axial position in the mill, except right at the bottom of the mill

    0.0

    2.0

    4.0

    0 0.5 1

    Height (m)

    RateofNormEAbs(

    mW)

    0.0

    2.0

    4.0

    0 0.05 0.1 0.15

    Mill Radius (m)

    RateofNormEAbs(

    mW)

    -5.0

    0.0

    5.0

    10.0

    15.0

    20.0

    0 0.5 1

    Height (m)RateofShear

    EAbs(mW)

    0.0

    5.0

    10.0

    15.0

    20.0

    0 0.05 0.1 0.15

    Mill Radius (m)

    RateofShe

    arEAbs(mW)

    0.0

    5.0

    10.0

    15.0

    20.0

    0 0.5 1

    Height (m)

    RateofTotalEAbs(mW)

    0.0

    5.0

    10.0

    15.0

    20.0

    0 0.05 0.1 0.15

    Mill Radius (m)

    RateofTotalEAbs(mW)

    Figure 13. One-dimensional profiles of media energy absorption rates versus mill heightand radius for the tower mill collected over only two revolutions when in steady state.Confidence limits of one standard deviation show the variance in velocity. The radius of

    the screw is 0.07 m and the chamber radius is 0.12 m.

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    344 P. W. CLEARY, M. D. SINNOTT AND R. D. MORRISON

    where they are lower because the screw impeller does not reach down to the floor of the mill. The

    energy absorption by the media declines reasonably strongly over the top third of the mill. The

    shear energy absorption in the bulk of the mill also shows a modest increasing trend with increasing

    height. The mean energy absorption rates all have strong radial dependence. They are relatively

    low at small radii near the shaft of the screw since the centrifugal forces are smaller here. Theenergy absorption rates increase rapidly with radii and reach peak values at about 0.085 m, which

    is moderately outside the outer diameter of the screw and well beyond the location of the peak

    axial velocity. This is the region of highest shear between the upward- and downward-directed

    streams of particles. Energy absorption then decreases with increasing radius but is still significant

    all the way out to the mill chamber wall.

    The average power draw for the tower mill was predicted to be 0.27 kW. For more details of

    the media motion and energy consumption in this tower mill see [31] and for details of mixing

    and transport [32].

    5.2. Isamill

    The Isamill is a horizontal stirred mill with multiple grinding discs on a central rotating shaft.

    Since it has a closed grinding vessel and does not rely on gravity to keep the mill charge within

    the mill, significantly higher rotational speeds can be used for this mill. Speed increases of factors

    of 1020 are possible. The much higher speeds produce centrifugal forces that are 100400 times

    higher, leading to much higher pressures, much higher shear rates and much higher levels of energy

    consumption for breakage. This type of mill can be used for ultra-fine grinding with product sizes

    down to a few microns possible. These very fine products can often be produced at favourable

    energy consumption rates if well tuned to the material being ground. The stirring agitator used is

    a shaft with several discs attached at regular intervals along the shaft. These extend most of the

    way to the outer cylindrical shell. Typically the discs have 46 holes (passing completely through

    the disc) with circular or elliptical shapes regularly spaced around the discs at about half the

    radius of the disc. The mill is filled to 75% volume with a charge composed mostly of grindingmedia (which could be steel or ceramic balls, sand or fine slag). The discs partition the charge

    into distinct zones or compartments and the disc rotation generates recirculating motion of the

    grinding media within and between the compartments. Fine feed material is added at one end

    under high pressure and is transported along the mill, being progressively ground by the media

    and then ideally discharged at the other end. The media is retained in the mill by a centrifugal

    separator.

    Here, we model just one compartment around one of the discs of a laboratory scale Isamill

    using periodic boundary conditions in the axial direction. The mill details are given in Table VIII.

    The mill has a diameter of 337.5 mm and the periodic slice is 200 mm (being the centre to centre

    spacing of the discs along the shaft). Details of the grinding media and the feed ore are also given

    in Table VIII. We wish to understand the distribution of ore within the charge so feed particles need

    to be included in the simulation. The feed ore is represented as 1 mm particles which are over-sizedcompared with the actual 0.2 mm feed of interest. Since these particles are small compared with

    the 6 mm grinding media, this is expected to give reasonable predictions for the spatial distribution

    of feed. We would not expect this approach to be reasonable for predicting the breakage of the

    feed particles by the media.

    Figure 14 shows a side view of the periodic slice of the Isamill. The mill is sectioned to allow

    viewing inside the charge. The 6 mm media are shown as blue and the 1 mm ore particles are shown

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 345

    Table VIII. Isamill configuration, charge and operating conditions.

    Mill shell radius 168.75 mmGrinding discs 145 mm radius disk

    25 mm thick30% cross-sectional area open to flow

    (provided by 4 circular holes)Shaft diameter 25 mm

    Grinding media 6 mm balls with density 2700 kg/m3

    Media volume of the charge 80% v/v

    Ore feed 1 mm and 5100 kg/m3

    Ore fraction of the charge 20% v/vCharge volume fill 75%Shaft/disc speed 1315 rpmDisc tip speed 20 m/s

    Figure 14. Media (blue) and fine feed material (red) distributions at steady state in an Isamill. The feedmaterial segregates rapidly to the outside filling the interstitial gaps between the media particles.

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    346 P. W. CLEARY, M. D. SINNOTT AND R. D. MORRISON

    Figure 15. Grinding media (blue) and feed ore (red) distributions in different slices of the Isamill, showing:(a) slice at the feed end just before the grinding disk; (b) in a slice aligned with the grinding disk; and

    (c) in a slice on the discharge side of the grinding disk.

    as red. The pressure gradient on the charge (implemented as an axial body force) pushes particles

    against the left side of the disc and leaves a matching void on the right side. The media densely

    packs the mill in all locations except in this void. The feed material distribution is more complex.

    The centrifugal force is extremely high and pushes these particles outwards. Since the feed is so

    small compared with the media, it is relatively free to migrate through the void spaces betweenthe close-packed media particles until all the pores are densely filled, starting from the mill shell

    and moving inwards. Once in steady state, the distance that the ore particles penetrate in towards

    the axis of the mill is then determined by the feed ore volume fraction. A full mill can be pictured

    by placing several copies of this periodic slice along the axial direction. The periodic nature of

    the computational domain does not require the spatial distribution of charge to be uniform in the

    axial direction within an individual periodic region. The distribution along the mill is periodic (not

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 347

    uniform) with successful dense zones on the feed sides and void zones on the discharge sides of

    each of the grinding discs.

    Figure 15 shows axial slices of the mill charge at three locations along the axis of the mill. The

    first one lies in the feed side of the mill compartment and shows densely packed media reaching

    in to the location of the shaft. The ore densely fills the voids around the outside of the mill andextends inwards for a reasonable distance. The interface is not smooth at the edge of the ore since

    the recirculation of the media (which is comparatively slow compared with the speed of the fines

    migration) causes some mixing in the interface region. The second slice is at the location of the

    disc. The media and ore are both confined to the narrow annulus around the outside of the disk.

    The locations of the four circular holes in the disc can be identified by the presence of some media

    and ore particles that are currently located in the openings, being moved around circumferentially

    by the disk. The third slice shows the charge distribution part way along the discharge side of the

    periodic volume. The media still densely fills the space between the shaft and the shell, but the

    amount of feed material is much reduced because of the restriction to axial transport that the disc

    represents. The ore particles are again densely packed from the shell inwards but now extend only

    about half the distance towards the shaft. The interface is also more diffuse than on the feed side.

    It is also useful to understand the nature of the particle motion in different regions of the mill

    charge. To do this, Figure 16 shows individual trajectories for selected media and rock particles.

    The trajectory thickness is drawn to match the diameter of the particle (hence, thick curves are for

    media particles and thin curves are for the fine rock particles). Figure 16(a) shows a media particle

    between discs at a moderate radius from the shaft. This particle executes close to circular motion

    with some axial motion (which appears to be of the random walk type rather than a coherent axial

    flow). Figure 16(c) shows a similar trajectory for a particle at a large radius near the outer shell.

    It again follows an almost circular path. Figure 16(e) shows a media particle at a similar initial

    radius but starting closer to the disc. The particle follows a clear spiral pattern, moving with a

    reasonably steady axial speed past the grinding disc and into the next compartment between discs.

    Figure 16(b) shows a feed particle starting in a similar location to the media particle in Figure

    16(e). It also moves in a coherent spiral pattern past the outside of the grinding disc into the nextcompartment. This shows that there is a coherent axial flow along the mill at radii larger than the

    discs. Figure 16(d) shows another trajectory for a feed particle that starts at a smaller radius. This

    particle moves in a spiral pattern but with increasing radius. This demonstrates that the axial flow

    extends some distance into the space between discs and that particles are able to migrate outward

    and then spiral past the grinding disk. Figure 16(f) shows a feed particle that starts at a similar

    radius but further from the disk. This particle travels on an essentially closed circular path with

    small random walk variations in the axial and radial directions. Many cycles of the rotation are

    shown to establish that the path is a stable closed trajectory.

    Figure 17 shows multiple trajectories of some other media and rock particles within the Isamill.

    In the top picture, one media and one feed particle are shown together. The media particle lies on

    a stable closed trajectory, whereas the feed particle trajectory spirals in the axial direction with the

    radius of the circulation decreasing once the particle has passed the centre disc and finally passingthrough one of the holes in the next grinding disk to the left. This demonstrates that small changes

    in the particle position can lead to significant differences in the trajectories and also demonstrates

    the flow of particles through the grinding disc holes. Figure 17(b) shows some additional media

    trajectories that spiral past the outside of the grinding disk. The range of variation of the particle

    trajectories is substantial and occurs for only small changes in the particle starting positions. This

    demonstrates that the velocity field for the charge is spatially complex and presents significant

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    348 P. W. CLEARY, M. D. SINNOTT AND R. D. MORRISON

    Figure 16. Selected individual trajectories of particles within the Isamill (left) mediatrajectories and (right) rock trajectories.

    challenges for visualization and interpretation. Trajectories allow the phase space to be sampled

    but they are not suitable for detailed analysis of the entire velocity field. It is not clear at this point

    if the velocity field is steady or if there are important transient variations associated with these

    various types of motion. If the velocity field is steady then spatially averaging the velocities ontoan Eulerian grid is one approach that can be taken to help in analysing the velocity field.

    Figure 18 shows the energy spectra for this Isamill configuration. They appear to be skewed

    normal distributions. The collision energies range from 107 up to 2000 J. Compared with the

    ball mill (in Figure 5) the peak energy levels are increased by a factor 200, demonstrating the

    significant increase in energy intensity obtainable for such high super-critical mill rotation speeds.

    The minimum collision energy tail also extends down to much lower energies (107 J), indicating

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    DEM PREDICTION OF PARTICLE FLOWS IN GRINDING PROCESSES 349

    Figure 17. Multiple selected trajectories of media and rock particles within the Isamill.

    that in regions of the mill the conditions are relatively benign with many very weak collisions

    occurring.

    Considering the rockrock energy spectra, we find that for the shear energy, dissipation has a

    similar shape and energy dissipation level to that of the overall spectra, but significantly reduced

    normal impact energy levels (by a factor of around 200). In contrast, the normal and shear energy

    absorption spectra are quite similar for the mediarock collisions. Looking carefully at the normal

    energy spectra for all collisions, we observe a clear double peak. From the sub-spectra, it is obvious

    that the first peak arises from the rockrock collisions and the second higher energy peak arises

    from the mediarock collisions. The rockliner spectra cover a similar range of energies but the

    distributions are more symmetric with the modal peak moving from around 1 J down to around

    0.5 J. The frequency of the highest energy collisions also decreases from around 600 collisions/sto only 10 collisions/s, leading to more than a